Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mapdh6dN Structured version   Unicode version

Theorem mapdh6dN 35692
Description: Lemmma for mapdh6N 35700. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
Hypotheses
Ref Expression
mapdh.q  |-  Q  =  ( 0g `  C
)
mapdh.i  |-  I  =  ( x  e.  _V  |->  if ( ( 2nd `  x
)  =  .0.  ,  Q ,  ( iota_ h  e.  D  ( ( M `  ( N `
 { ( 2nd `  x ) } ) )  =  ( J `
 { h }
)  /\  ( M `  ( N `  {
( ( 1st `  ( 1st `  x ) ) 
.-  ( 2nd `  x
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } ) ) ) ) )
mapdh.h  |-  H  =  ( LHyp `  K
)
mapdh.m  |-  M  =  ( (mapd `  K
) `  W )
mapdh.u  |-  U  =  ( ( DVecH `  K
) `  W )
mapdh.v  |-  V  =  ( Base `  U
)
mapdh.s  |-  .-  =  ( -g `  U )
mapdhc.o  |-  .0.  =  ( 0g `  U )
mapdh.n  |-  N  =  ( LSpan `  U )
mapdh.c  |-  C  =  ( (LCDual `  K
) `  W )
mapdh.d  |-  D  =  ( Base `  C
)
mapdh.r  |-  R  =  ( -g `  C
)
mapdh.j  |-  J  =  ( LSpan `  C )
mapdh.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
mapdhc.f  |-  ( ph  ->  F  e.  D )
mapdh.mn  |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( J `  { F } ) )
mapdhcl.x  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
mapdh.p  |-  .+  =  ( +g  `  U )
mapdh.a  |-  .+b  =  ( +g  `  C )
mapdh6d.xn  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )
mapdh6d.yz  |-  ( ph  ->  ( N `  { Y } )  =  ( N `  { Z } ) )
mapdh6d.y  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
mapdh6d.z  |-  ( ph  ->  Z  e.  ( V 
\  {  .0.  }
) )
mapdh6d.w  |-  ( ph  ->  w  e.  ( V 
\  {  .0.  }
) )
mapdh6d.wn  |-  ( ph  ->  -.  w  e.  ( N `  { X ,  Y } ) )
Assertion
Ref Expression
mapdh6dN  |-  ( ph  ->  ( I `  <. X ,  F ,  ( w  .+  ( Y 
.+  Z ) )
>. )  =  (
( I `  <. X ,  F ,  w >. )  .+b  ( I `  <. X ,  F ,  ( Y  .+  Z ) >. )
) )
Distinct variable groups:    x, D, h    h, F, x    x, J    x, M    x, N    x,  .0.    x, Q    x, R    x, 
.-    h, X, x    h, Y, x    ph, h    .0. , h    C, h    D, h   
h, J    h, M    h, N    R, h    U, h    .- , h    w, h    h, Z, x    .+b , h    h, I, x    .+ , h, x   
x, w
Allowed substitution hints:    ph( x, w)    C( x, w)    D( w)    .+ ( w)    .+b ( x, w)    Q( w, h)    R( w)    U( x, w)    F( w)    H( x, w, h)    I( w)    J( w)    K( x, w, h)    M( w)    .- ( w)    N( w)    V( x, w, h)    W( x, w, h)    X( w)    Y( w)    .0. ( w)    Z( w)

Proof of Theorem mapdh6dN
StepHypRef Expression
1 mapdh.h . . . . . 6  |-  H  =  ( LHyp `  K
)
2 mapdh.c . . . . . 6  |-  C  =  ( (LCDual `  K
) `  W )
3 mapdh.k . . . . . 6  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
41, 2, 3lcdlmod 35545 . . . . 5  |-  ( ph  ->  C  e.  LMod )
5 mapdh.q . . . . . 6  |-  Q  =  ( 0g `  C
)
6 mapdh.i . . . . . 6  |-  I  =  ( x  e.  _V  |->  if ( ( 2nd `  x
)  =  .0.  ,  Q ,  ( iota_ h  e.  D  ( ( M `  ( N `
 { ( 2nd `  x ) } ) )  =  ( J `
 { h }
)  /\  ( M `  ( N `  {
( ( 1st `  ( 1st `  x ) ) 
.-  ( 2nd `  x
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } ) ) ) ) )
7 mapdh.m . . . . . 6  |-  M  =  ( (mapd `  K
) `  W )
8 mapdh.u . . . . . 6  |-  U  =  ( ( DVecH `  K
) `  W )
9 mapdh.v . . . . . 6  |-  V  =  ( Base `  U
)
10 mapdh.s . . . . . 6  |-  .-  =  ( -g `  U )
11 mapdhc.o . . . . . 6  |-  .0.  =  ( 0g `  U )
12 mapdh.n . . . . . 6  |-  N  =  ( LSpan `  U )
13 mapdh.d . . . . . 6  |-  D  =  ( Base `  C
)
14 mapdh.r . . . . . 6  |-  R  =  ( -g `  C
)
15 mapdh.j . . . . . 6  |-  J  =  ( LSpan `  C )
16 mapdhc.f . . . . . 6  |-  ( ph  ->  F  e.  D )
17 mapdh.mn . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( J `  { F } ) )
18 mapdhcl.x . . . . . 6  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
19 mapdh6d.w . . . . . . 7  |-  ( ph  ->  w  e.  ( V 
\  {  .0.  }
) )
2019eldifad 3440 . . . . . 6  |-  ( ph  ->  w  e.  V )
211, 8, 3dvhlvec 35062 . . . . . . . . 9  |-  ( ph  ->  U  e.  LVec )
2218eldifad 3440 . . . . . . . . 9  |-  ( ph  ->  X  e.  V )
23 mapdh6d.y . . . . . . . . . 10  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
2423eldifad 3440 . . . . . . . . 9  |-  ( ph  ->  Y  e.  V )
25 mapdh6d.wn . . . . . . . . 9  |-  ( ph  ->  -.  w  e.  ( N `  { X ,  Y } ) )
269, 12, 21, 20, 22, 24, 25lspindpi 17321 . . . . . . . 8  |-  ( ph  ->  ( ( N `  { w } )  =/=  ( N `  { X } )  /\  ( N `  { w } )  =/=  ( N `  { Y } ) ) )
2726simpld 459 . . . . . . 7  |-  ( ph  ->  ( N `  {
w } )  =/=  ( N `  { X } ) )
2827necomd 2719 . . . . . 6  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { w } ) )
295, 6, 1, 7, 8, 9, 10, 11, 12, 2, 13, 14, 15, 3, 16, 17, 18, 20, 28mapdhcl 35680 . . . . 5  |-  ( ph  ->  ( I `  <. X ,  F ,  w >. )  e.  D )
30 mapdh.a . . . . . 6  |-  .+b  =  ( +g  `  C )
3113, 30, 5lmod0vrid 17087 . . . . 5  |-  ( ( C  e.  LMod  /\  (
I `  <. X ,  F ,  w >. )  e.  D )  -> 
( ( I `  <. X ,  F ,  w >. )  .+b  Q
)  =  ( I `
 <. X ,  F ,  w >. ) )
324, 29, 31syl2anc 661 . . . 4  |-  ( ph  ->  ( ( I `  <. X ,  F ,  w >. )  .+b  Q
)  =  ( I `
 <. X ,  F ,  w >. ) )
3332adantr 465 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =  .0.  )  ->  ( (
I `  <. X ,  F ,  w >. ) 
.+b  Q )  =  ( I `  <. X ,  F ,  w >. ) )
34 oteq3 4170 . . . . . 6  |-  ( ( Y  .+  Z )  =  .0.  ->  <. X ,  F ,  ( Y  .+  Z ) >.  =  <. X ,  F ,  .0.  >.
)
3534fveq2d 5795 . . . . 5  |-  ( ( Y  .+  Z )  =  .0.  ->  (
I `  <. X ,  F ,  ( Y  .+  Z ) >. )  =  ( I `  <. X ,  F ,  .0.  >. ) )
365, 6, 11, 18, 16mapdhval0 35678 . . . . 5  |-  ( ph  ->  ( I `  <. X ,  F ,  .0.  >.
)  =  Q )
3735, 36sylan9eqr 2514 . . . 4  |-  ( (
ph  /\  ( Y  .+  Z )  =  .0.  )  ->  ( I `  <. X ,  F ,  ( Y  .+  Z ) >. )  =  Q )
3837oveq2d 6208 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =  .0.  )  ->  ( (
I `  <. X ,  F ,  w >. ) 
.+b  ( I `  <. X ,  F , 
( Y  .+  Z
) >. ) )  =  ( ( I `  <. X ,  F ,  w >. )  .+b  Q
) )
39 oveq2 6200 . . . . . 6  |-  ( ( Y  .+  Z )  =  .0.  ->  (
w  .+  ( Y  .+  Z ) )  =  ( w  .+  .0.  ) )
401, 8, 3dvhlmod 35063 . . . . . . 7  |-  ( ph  ->  U  e.  LMod )
41 mapdh.p . . . . . . . 8  |-  .+  =  ( +g  `  U )
429, 41, 11lmod0vrid 17087 . . . . . . 7  |-  ( ( U  e.  LMod  /\  w  e.  V )  ->  (
w  .+  .0.  )  =  w )
4340, 20, 42syl2anc 661 . . . . . 6  |-  ( ph  ->  ( w  .+  .0.  )  =  w )
4439, 43sylan9eqr 2514 . . . . 5  |-  ( (
ph  /\  ( Y  .+  Z )  =  .0.  )  ->  ( w  .+  ( Y  .+  Z
) )  =  w )
4544oteq3d 4173 . . . 4  |-  ( (
ph  /\  ( Y  .+  Z )  =  .0.  )  ->  <. X ,  F ,  ( w  .+  ( Y  .+  Z
) ) >.  =  <. X ,  F ,  w >. )
4645fveq2d 5795 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =  .0.  )  ->  ( I `  <. X ,  F ,  ( w  .+  ( Y  .+  Z ) ) >. )  =  ( I `  <. X ,  F ,  w >. ) )
4733, 38, 463eqtr4rd 2503 . 2  |-  ( (
ph  /\  ( Y  .+  Z )  =  .0.  )  ->  ( I `  <. X ,  F ,  ( w  .+  ( Y  .+  Z ) ) >. )  =  ( ( I `  <. X ,  F ,  w >. )  .+b  ( I `  <. X ,  F ,  ( Y  .+  Z ) >. )
) )
483adantr 465 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( K  e.  HL  /\  W  e.  H ) )
4916adantr 465 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  F  e.  D
)
5017adantr 465 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( M `  ( N `  { X } ) )  =  ( J `  { F } ) )
5118adantr 465 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  X  e.  ( V  \  {  .0.  } ) )
5219adantr 465 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  w  e.  ( V  \  {  .0.  } ) )
53 mapdh6d.z . . . . . . 7  |-  ( ph  ->  Z  e.  ( V 
\  {  .0.  }
) )
5453eldifad 3440 . . . . . 6  |-  ( ph  ->  Z  e.  V )
559, 41lmodvacl 17070 . . . . . 6  |-  ( ( U  e.  LMod  /\  Y  e.  V  /\  Z  e.  V )  ->  ( Y  .+  Z )  e.  V )
5640, 24, 54, 55syl3anc 1219 . . . . 5  |-  ( ph  ->  ( Y  .+  Z
)  e.  V )
5756anim1i 568 . . . 4  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( ( Y 
.+  Z )  e.  V  /\  ( Y 
.+  Z )  =/= 
.0.  ) )
58 eldifsn 4100 . . . 4  |-  ( ( Y  .+  Z )  e.  ( V  \  {  .0.  } )  <->  ( ( Y  .+  Z )  e.  V  /\  ( Y 
.+  Z )  =/= 
.0.  ) )
5957, 58sylibr 212 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( Y  .+  Z )  e.  ( V  \  {  .0.  } ) )
60 mapdh6d.yz . . . . . . 7  |-  ( ph  ->  ( N `  { Y } )  =  ( N `  { Z } ) )
61 mapdh6d.xn . . . . . . . . 9  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )
629, 12, 21, 22, 24, 54, 61lspindpi 17321 . . . . . . . 8  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Y } )  /\  ( N `  { X } )  =/=  ( N `  { Z } ) ) )
6362simpld 459 . . . . . . 7  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
649, 41, 11, 12, 21, 18, 23, 53, 19, 60, 63, 25mapdindp1 35673 . . . . . 6  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { ( Y  .+  Z ) } ) )
659, 41, 11, 12, 21, 18, 23, 53, 19, 60, 63, 25mapdindp2 35674 . . . . . 6  |-  ( ph  ->  -.  w  e.  ( N `  { X ,  ( Y  .+  Z ) } ) )
669, 11, 12, 21, 18, 56, 20, 64, 65lspindp1 17322 . . . . 5  |-  ( ph  ->  ( ( N `  { w } )  =/=  ( N `  { ( Y  .+  Z ) } )  /\  -.  X  e.  ( N `  {
w ,  ( Y 
.+  Z ) } ) ) )
6766simprd 463 . . . 4  |-  ( ph  ->  -.  X  e.  ( N `  { w ,  ( Y  .+  Z ) } ) )
6867adantr 465 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  -.  X  e.  ( N `  { w ,  ( Y  .+  Z ) } ) )
6926simprd 463 . . . . . . . . 9  |-  ( ph  ->  ( N `  {
w } )  =/=  ( N `  { Y } ) )
709, 11, 12, 21, 19, 24, 69lspsnne1 17306 . . . . . . . 8  |-  ( ph  ->  -.  w  e.  ( N `  { Y } ) )
71 eqid 2451 . . . . . . . . . 10  |-  ( LSSum `  U )  =  (
LSSum `  U )
729, 12, 71, 40, 24, 54lsmpr 17278 . . . . . . . . 9  |-  ( ph  ->  ( N `  { Y ,  Z }
)  =  ( ( N `  { Y } ) ( LSSum `  U ) ( N `
 { Z }
) ) )
7360oveq2d 6208 . . . . . . . . 9  |-  ( ph  ->  ( ( N `  { Y } ) (
LSSum `  U ) ( N `  { Y } ) )  =  ( ( N `  { Y } ) (
LSSum `  U ) ( N `  { Z } ) ) )
74 eqid 2451 . . . . . . . . . . . . 13  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
759, 74, 12lspsncl 17166 . . . . . . . . . . . 12  |-  ( ( U  e.  LMod  /\  Y  e.  V )  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
7640, 24, 75syl2anc 661 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
7774lsssubg 17146 . . . . . . . . . . 11  |-  ( ( U  e.  LMod  /\  ( N `  { Y } )  e.  (
LSubSp `  U ) )  ->  ( N `  { Y } )  e.  (SubGrp `  U )
)
7840, 76, 77syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  ( N `  { Y } )  e.  (SubGrp `  U ) )
7971lsmidm 16267 . . . . . . . . . 10  |-  ( ( N `  { Y } )  e.  (SubGrp `  U )  ->  (
( N `  { Y } ) ( LSSum `  U ) ( N `
 { Y }
) )  =  ( N `  { Y } ) )
8078, 79syl 16 . . . . . . . . 9  |-  ( ph  ->  ( ( N `  { Y } ) (
LSSum `  U ) ( N `  { Y } ) )  =  ( N `  { Y } ) )
8172, 73, 803eqtr2d 2498 . . . . . . . 8  |-  ( ph  ->  ( N `  { Y ,  Z }
)  =  ( N `
 { Y }
) )
8270, 81neleqtrrd 2564 . . . . . . 7  |-  ( ph  ->  -.  w  e.  ( N `  { Y ,  Z } ) )
839, 41, 12, 40, 24, 54, 20, 82lspindp4 17326 . . . . . 6  |-  ( ph  ->  -.  w  e.  ( N `  { Y ,  ( Y  .+  Z ) } ) )
849, 12, 21, 20, 24, 56, 83lspindpi 17321 . . . . 5  |-  ( ph  ->  ( ( N `  { w } )  =/=  ( N `  { Y } )  /\  ( N `  { w } )  =/=  ( N `  { ( Y  .+  Z ) } ) ) )
8584simprd 463 . . . 4  |-  ( ph  ->  ( N `  {
w } )  =/=  ( N `  {
( Y  .+  Z
) } ) )
8685adantr 465 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( N `  { w } )  =/=  ( N `  { ( Y  .+  Z ) } ) )
87 eqidd 2452 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( I `  <. X ,  F ,  w >. )  =  ( I `  <. X ,  F ,  w >. ) )
88 eqidd 2452 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( I `  <. X ,  F , 
( Y  .+  Z
) >. )  =  ( I `  <. X ,  F ,  ( Y  .+  Z ) >. )
)
895, 6, 1, 7, 8, 9, 10, 11, 12, 2, 13, 14, 15, 48, 49, 50, 51, 41, 30, 52, 59, 68, 86, 87, 88mapdh6aN 35688 . 2  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( I `  <. X ,  F , 
( w  .+  ( Y  .+  Z ) )
>. )  =  (
( I `  <. X ,  F ,  w >. )  .+b  ( I `  <. X ,  F ,  ( Y  .+  Z ) >. )
) )
9047, 89pm2.61dane 2766 1  |-  ( ph  ->  ( I `  <. X ,  F ,  ( w  .+  ( Y 
.+  Z ) )
>. )  =  (
( I `  <. X ,  F ,  w >. )  .+b  ( I `  <. X ,  F ,  ( Y  .+  Z ) >. )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2644   _Vcvv 3070    \ cdif 3425   ifcif 3891   {csn 3977   {cpr 3979   <.cotp 3985    |-> cmpt 4450   ` cfv 5518   iota_crio 6152  (class class class)co 6192   1stc1st 6677   2ndc2nd 6678   Basecbs 14278   +g cplusg 14342   0gc0g 14482   -gcsg 15517  SubGrpcsubg 15779   LSSumclsm 16239   LModclmod 17056   LSubSpclss 17121   LSpanclspn 17160   HLchlt 33303   LHypclh 33936   DVecHcdvh 35031  LCDualclcd 35539  mapdcmpd 35577
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462  ax-riotaBAD 32912
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-ot 3986  df-uni 4192  df-int 4229  df-iun 4273  df-iin 4274  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-of 6422  df-om 6579  df-1st 6679  df-2nd 6680  df-tpos 6847  df-undef 6894  df-recs 6934  df-rdg 6968  df-1o 7022  df-oadd 7026  df-er 7203  df-map 7318  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-2 10483  df-3 10484  df-4 10485  df-5 10486  df-6 10487  df-n0 10683  df-z 10750  df-uz 10965  df-fz 11541  df-struct 14280  df-ndx 14281  df-slot 14282  df-base 14283  df-sets 14284  df-ress 14285  df-plusg 14355  df-mulr 14356  df-sca 14358  df-vsca 14359  df-0g 14484  df-mre 14628  df-mrc 14629  df-acs 14631  df-poset 15220  df-plt 15232  df-lub 15248  df-glb 15249  df-join 15250  df-meet 15251  df-p0 15313  df-p1 15314  df-lat 15320  df-clat 15382  df-mnd 15519  df-submnd 15569  df-grp 15649  df-minusg 15650  df-sbg 15651  df-subg 15782  df-cntz 15939  df-oppg 15965  df-lsm 16241  df-cmn 16385  df-abl 16386  df-mgp 16699  df-ur 16711  df-rng 16755  df-oppr 16823  df-dvdsr 16841  df-unit 16842  df-invr 16872  df-dvr 16883  df-drng 16942  df-lmod 17058  df-lss 17122  df-lsp 17161  df-lvec 17292  df-lsatoms 32929  df-lshyp 32930  df-lcv 32972  df-lfl 33011  df-lkr 33039  df-ldual 33077  df-oposet 33129  df-ol 33131  df-oml 33132  df-covers 33219  df-ats 33220  df-atl 33251  df-cvlat 33275  df-hlat 33304  df-llines 33450  df-lplanes 33451  df-lvols 33452  df-lines 33453  df-psubsp 33455  df-pmap 33456  df-padd 33748  df-lhyp 33940  df-laut 33941  df-ldil 34056  df-ltrn 34057  df-trl 34111  df-tgrp 34695  df-tendo 34707  df-edring 34709  df-dveca 34955  df-disoa 34982  df-dvech 35032  df-dib 35092  df-dic 35126  df-dih 35182  df-doch 35301  df-djh 35348  df-lcdual 35540  df-mapd 35578
This theorem is referenced by:  mapdh6gN  35695
  Copyright terms: Public domain W3C validator